Fully invariant and verbal congruence relations

A congruence relation theta on an algebra A is fully invariant if every endomorphism of A preserves theta. A congruence theta is verbal if there exists a variety such that theta is the least congruence of A such that . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.